An algorithmic discrete gradient field for non-colliding cell-like objects and the topology of pairs of points on skeleta of simplexes

For a positive integer $n$ and a finite simplicial complex $K$, we describe an algorithmic procedure constructing a maximal discrete gradient field $W(K,n)$ on Abrams’ discretized configuration space $\text{DConf}(K,n)$. Computer experimentation shows that the field is generically optimal. We study the field $W(K,n)$ for $n=2$ and $K={\mathrm{\Delta }}^{m,d}$, the $d$-dimensional skeleton of the $m$-dimensional simplex. In particular, we prove that $\text{DConf}({\mathrm{\Delta }}^{m,d},2)$ is $(\min \{d,m-1\}-1)$-connected, has torsion-free homology and admits a minimal cell structure. We compute the Betti numbers of $\text{DConf}({\mathrm{\Delta }}^{m,d},2)$ and, for certain values of $d$, we prove that $\text{DConf}({\mathrm{\Delta }}^{m,d},2)$ breaks, up to homotopy, as a wedge of (not necessarily equidimensional) spheres.